We consider the isoperimetric problem, that is the minimization of the measure of the boundary among subsets having a given volume, on noncompact manifolds with nonnegative Ricci curvature, Euclidean volume growth and with quadratic Riemann curvature decay. The aim of the talk is to discuss uniqueness and stability properties of minimizers, called isoperimetric sets. Assuming the manifold is not the Euclidean space, we show that for most large volumes (in a quantified way) there exists a unique isoperimetric set, and its boundary is strictly volume preserving stable. Uniqueness here is meant in the set theoretical sense and it is not understood up to isometry of the ambient. We show with a counterexample that the result cannot be improved to uniqueness or strict stability for every large volume. The lack of higher regularity at infinity prevents the application of classical methods based on the implicit function theorem. A key tool for deriving the needed effective estimates on the Jacobi operator of the boundary of large isoperimetric sets is provided here by the sharp concavity property of the isoperimetric profile function, which will be briefly reviewed. The talk is based on a joint work with Gioacchino Antonelli and Daniele Semola.